1. When we start English study, what is the first thing that we must study conversation, composition, or grammar? Which will help us the most when we prepare for TOEIC? Why?

When it comes to studying English, everyone seems to have a hard time deciding where to start; Conversation, Writing, or Grammar. This might be because each area is so much interrelated that if we abandon one area in order to study another area first, we will certainly encounter difficulties in advancing to a higher level. Therefore, we had better study each area at the same time!
If there is some INPUT, there should be some OUTPUT. For example, we can take Grammar as INPUT. Most students tend to stick to studying Grammar, but Grammar is not an end in itself. It should be a means to an end: making ourselves understood through communication. Let's imagine that we have learned something about the subjunctive mood (INPUT). Then, we need to speak and write down something using the subjunctive mood (OUTPUT). In that way, we can study each area at the same time.
The same can be true of TOEIC, too. If we don't know some specific expression such as "If Subject had p.p. Subject would/could/might have p.p." we may not understand it when we hear it in parts 1 through 4. In addition, there might be a case we need to use that expression in TOEIC speaking and writing tests. To sum up, we cannot say which area is more helpful or important. As noted earlier, each area should be dealt with at the same time.

2. What do you suggest about learning composition?

A nicely organized (or well-written) passage is usually composed of several paragraphs. Then, each of these paragraphs consists of sentences free of grammatical errors. In other words, a good paragraph builds on good sentences, and a good passage on good paragraphs. From this fact, it naturally follows that we should start from writing a good sentence.
Here, studying grammar plays a role. Grammar is not important in itself, but it can help us produce a grammatically correct sentence. Grammar should be studied in such a way that learners themselves realize how a specific grammatical expression is used in producing a good sentence. So, when we study grammar, we should make a habit of trying to write sentences using grammatical expressions we have learned. Then, we move on to the next step: how to write a paragraph.
To come up with a good paragraph, we need to have an eye for a good paragraph, which means it is important to be able to tell a well-structured paragraph from a bad one. Hence, reading others' composition should precede writing a paragraph. By doing it this way, we can come to know how a paragraph should be structured to draw readers' attention and make our writing more convincing.

3. How can we become good at English listening?

There is no established rule for practicing listening comprehension. But one of the most important things is that we need to be exposed to a learning environment where we can be immersed in English listening as much as we can. In other words, it is important we have as many chances to listen to English as possible. In doing so, there are some recommended steps to take. After listening to a sentence spoken by a native speaker, try to repeat after him or her without referring to the script. Just try to focus on what you hear through your ears. If you look at the script, the information you want to get will be processed though your eyes. In listening comprehension, the information you want to get should be processed through your ears. While repeating what you hear several times, you will be happy to hear words that you couldn't hear at first. After the process of repeating, take a look at the script, and then highlight words or expressions you haven't heard correctly. Then, close the script, and try to listen to the parts you have missed. Through this whole process, you are getting accustomed to words or expressions (you don't know) through your ears.
Next, when you study reading comprehension, try to highlight words or expressions (especially collocations) that you don't know. Then, get accustomed to how those words or expressions are pronounced. Getting accustomed to words or expressions through your ears is important because words can be pronounced differently when they are combined with other words or expressions. Lastly, do what is recommended here everyday, even for a short time.

4. When we try to write in English, sometimes we use ‘by’ and ‘that’ more than twice in one sentence. Is it OK?

It's OK to use the same words. However, you'd better be concise as best you can. Try not to use a word if deleting it doesn't cause any changes in meaning. Let's take a sentence as an example: "He admitted that he knew that she had killed his son." This sentence is grammatically correct, but somewhat redundant because of the repeated use of "that." It would be better to delete one 'that': "He admitted that he knew she had killed his son." Remember that even if the repeated use of a word is grammatically right, it may lead to redundancy.